On differential operators of numerical semigroup rings

If S=〈d1,…,dν〉 is a numerical semigroup, we call the ring C[S]=C[td1,…,tdν] the semigroup ring of S. We study the ring of differential operators on C[S], and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe if S is a so-called Arf semigroup. If I is an ideal in C[S] that is generated by monomials, we also give some results on (the set of derivations which map I into I).